STAT7055 Chap.7 Hypothesis Tests and Statistical Decisions

Hypothesis Tests and Statistical Decisions

Define null and alternative hypotheses

The captured teaching materials give this chapter a concrete anchor: A house-price example tests a benchmark near $400,000 and evaluates Type II error and power when the true mean is lower, so non-rejection is tied to a specified alternative rather than treated as proof.

That null and alternative hypotheses anchor controls how test statistic is explained and how p-value and decision is tested in changed practice.

Hypothesis Tests and Statistical Decisions is a quantitative decision problem built from null and alternative hypotheses, test statistic and p-value and decision.

The aim is to state hypotheses, evaluate compatibility with the null model and separate the statistical decision from the business conclusion; a numerical result earns meaning only when the variables, units, assumptions and comparison are all explicit.

Begin with null and alternative hypotheses: state what quantity it represents, the scale on which it is measured and the condition under which it changes.

Then map every symbol in the Hypothesis Tests and Statistical Decisions formula checkpoint to null and alternative hypotheses before calculation begins.

Next connect test statistic to the calculation. Show the test statistic transformation line by line, preserve units and signs, and make any denominator or baseline visible.

A test statistic calculator output is not a method; the reader must be able to reconstruct why that operation answers the question.

Trace test statistic

Use p-value and decision to interpret or stress-test the result.

Ask whether the p-value and decision magnitude is plausible, whether a boundary case behaves as expected and which conclusion would reverse if an assumption changed.

This is where computation becomes analysis rather than arithmetic.

When the task is to state hypotheses, evaluate compatibility with the null model and separate the statistical decision from the business conclusion, separate inputs supplied by the problem from quantities you derive.

Then report the p-value and decision result in the language of the course and attach the relevant uncertainty, limitation or decision consequence.

Build a representation check before solving.

Put null and alternative hypotheses, test statistic and p-value and decision into a small symbol-and-units table, mark which values are observed and which are calculated, and predict the direction of the result before doing arithmetic.

An null and alternative hypotheses sign, scale or unit mismatch then becomes visible at setup instead of being hidden inside a polished final number.

Run one sensitivity test after the baseline answer. Change the input most closely connected to test statistic, hold the remaining assumptions fixed and recompute only the affected steps. Explain whether the movement in p-value and decision matches the mechanism.

This test statistic sensitivity shows which assumption controls the conclusion and prevents a single scenario from being presented as universal.

Test with p-value and decision

Use a three-column null and alternative hypotheses error log for STAT7055: translation error, calculation error and interpretation error.

Record the exact line where the test statistic solution first diverged, rewrite that line, and check it with a limiting case or an independent calculation.

Correcting the first failed test statistic move is more useful than copying the complete solution again.

A complete response should make the task visible before the detail: identify what must be decided, define the relevant terms, connect the evidence to test statistic, and use p-value and decision to test the result.

The final sentence about p-value and decision should answer the question actually asked rather than merely repeat the topic.

The controlling limit is specific: A small p-value is not the probability that the null is true and does not measure practical importance.

Keep that p-value and decision limit beside the worked example, because it separates a careful STAT7055 answer from one that sounds confident but claims more than the task or evidence supports.

For revision, retrieve null and alternative hypotheses, test statistic and p-value and decision without notes, explain their relationship aloud, then complete a changed version of the application: state hypotheses, evaluate compatibility with the null model and separate the statistical decision from the business conclusion.

Record the first failed test statistic reasoning move and repair it before attempting another case.

What this chapter covers

• 01

  null and alternative hypotheses

• 02

  test statistic

• 03

  p-value and decision

• 04

  Applying null and alternative hypotheses

• 05

  Limits of test statistic and p-value and decision

AskSia practice: apply Hypothesis Tests and Statistical Decisions

Key terms

null and alternative hypotheses

Competing parameter claims in which the null gives a benchmark and the alternative specifies a departure. Use this definition when the task is to state hypotheses, evaluate compatibility with the null model and separate the statistical decision from the business conclusion.

test statistic

A sample quantity standardised under the null model to measure departure from its benchmark. Use this definition when the task is to state hypotheses, evaluate compatibility with the null model and separate the statistical decision from the business conclusion.

p-value and decision

The p-value measures extremeness under the null; a decision compares it with a prespecified level. Use this definition when the task is to state hypotheses, evaluate compatibility with the null model and separate the statistical decision from the business conclusion.

Hypothesis Tests and Statistical Decisions FAQ

Exam move

Reconstruct the relationship among null and alternative hypotheses, test statistic and p-value and decision; complete the chapter application without notes; then test the result against this limit: A small p-value is not the probability that the null is true and does not measure practical importance.

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